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In the process of implementing a project, you have sub- contracted JAB Enterprise to supply supplements worth 1480 cartons of two varieties X and Y. Suppose the joi...

Question

In the process of implementing a project, you have sub- contracted JAB Enterprise to supply supplements worth 1480 cartons of two varieties X and Y. Suppose the joint cost function for the two varieties of supplements is given by C = 250X2 + 120Y2. The quantity of X and Y are not specified; therefore, the firm may decide to supply any combination. The firm wishes to minimize the cost of producing the supplement but meet your demand. Using Lagrangian multiplier technique find the amounts of X a

In the process of implementing a project, you have sub- contracted JAB Enterprise to supply supplements worth 1480 cartons of two varieties X and Y. Suppose the joint cost function for the two varieties of supplements is given by C = 250X2 + 120Y2. The quantity of X and Y are not specified; therefore, the firm may decide to supply any combination. The firm wishes to minimize the cost of producing the supplement but meet your demand. Using Lagrangian multiplier technique find the amounts of X and Y that will minimize cost and compute this cost. Also examine the cost implications of changing this optimal combination so as to produce equal amounts of both products.



Answers

In the process of implementing a project, you have sub- contracted JAB Enterprise to supply supplements worth 1480 cartons of two varieties X and Y. Suppose the joint cost function for the two varieties of supplements is given by C = 250X2 + 120Y2.
The quantity of X and Y are not specified; therefore, the firm may decide to supply any combination. The firm wishes to minimize the cost of producing the supplement but meet your demand.
Using Lagrangian multiplier technique find the amounts of X and Y that will minimize cost and compute this cost. Also examine the cost implications of changing this optimal combination so as to produce equal amounts of both products.

For this problem. We've got a function sea of ex apply that gives us the cost of manufacturing items. For a company where the two variables X and Y X is the cost of labor per hour, why's the cost of materials per unit? And our goal is to find the values for X and Y that minimize our costs of manufacturing and find out what that cost actually is. So in order to do this, we first start by calculating our first order derivatives because we're looking for critical points. So the derivative with respect to X six ex wives a constant minus one minus three. Why this derivative with respect? Why X is a constant. So get to why minus one minus three X. And now we're going to evaluate for where points a comma Be that make these two expressions zero. So this 1st 1 implies that six a minus one minus three B equals zero we're gonna solve for a so a is equal to one over six times one plus three b. Our second expression is sea of why a comma be zero. This gives us to be minus one minus three a equal, sir. Now we know what a is in terms of bees. So we're gonna go ahead and put that right in here. This give me three over six one plus three B equal to zero. Just made her substitution for what is all right. And what we're gonna do is make things simpler for ourselves. We're gonna multiply both sides by six. We're gonna get 12 B minus six, minus three. Plus this stuff, we do that because fractions are awful and I don't want to do with fractions. So we multiplied everything by six. This is 12 B minus six, minus three minus nine. B equals zero. So we've got a total of three be on this side that is equal to nine. Because we moved the 33 and the six over, which gives us b is equal to three. Now we have our expression for what a is. It's one over six, one plus three b. But we know that B is three. So this is one over six times one plus nine. So that's 10 over six or I'm sorry or five over three. So a critical point is 5/3. Come on, three. Now we want to calculate our discriminative to figure out whether or not this is a maximum. Because if we maximize the cost of manufacturing, we'll get fired. So we probably shouldn't do that. So let's find out, uh, what the actual minimum is. We make sure that this is a minimum. We're gonna need our original expression here for the first derivatives, because we're gonna calculate Second Dorados. Now, these air, these functions f so we can clear our second derivatives. We take another derivative respect to X is gonna give us just a six. Another derivative with Inspector. Why, it's gonna give us sister T another derivative of the first expression. With respect to why now? This way. This right here is gonna be minus three. So are discriminated is six times too minus by this three squared. This is 12 minus nine. That's a positive three that's created in zero. So our discriminates better than zero. Our second derivative with stricter exes. So secretive and zero. So we found a minimum. Good. We're gonna keep our jobs. That's pretty nice. Now you want to find out exactly what that minimum is, so we're going to put into our cost formula are five over. Three comma or three. That's going to go. What? Well, I don't remember what the expression is. So we're gonna go ahead and go up here and take a copy of this. I'm gonna put that down here. There we go. And then we just have to plug in the values. So we get three times of 5/3 squared, plus a three squared minus 5/3 minus three minus three times, 5/3 times three lots of threes in this, plus 100. So this is gonna be equal to 25 over three. But if done there is, I just didn't 25 over nine. And then the multiple of three on the outside gives us just the three on the bottom plus nine minus 5/3 minus three minus 15. Because one of those threes cancels. So we got three times we go 100 So this is equal to 20 over three, plus, uh, nine and minus three is 66 minus 15. It's negative. Nine. So that's and so this is equal to two and 2/3. 06 and 2/3 plus 91. So this is 97 and 2/3 so the minimize costs is $97 and 67 sense

Party we want to know are fixed costs. Or that's just the cost to purchase a copier, which you know now for Part B. What's you know are variable costs. Well, that's going to be, um, amount a copy each, um, crosses around your pretty That's why don't you? Oh, this is each copying well, and then this is the price of a copier. Get an offer apart. See, we want to find our cost equation That's going to be C is equal to our price of our coffee are plus the mouth needed to call each are our amount of copy. So that's are now for 40. We want to find our calls for producing 10,000 copy. That's easy to do. West 0.2 times, I know that. Let's put that into our kid equation. So nothing. So we have 2200 and I'll report E. You want to find our X given nuts, we can produce a total of 2600. Okay, so it's not priced attracting. Here are the old science. We get 600 you get the point, dividing both sides by 0.2 x or just wake up to 600 divided backwards. You're too that's useful to There you go that soppy

So we're talking about the cost of producing items. And for number 80 the copier cost 2000 and it also costs two cents per copy to make items. And so for part A, this is what is the fixed cost. The fixed cost is the copier. That price is not changing. So $2000 is our fixed cost for part A for part B, what's the variable cost? Well, since the number of copies can change, that's your variable cost. So two cents 0.2 is the variable cost. And then for part C, it wants the cost equation. So the total cost is the variable cost times how many items there are, plus the fixed cost growth 2000. So there's a cost equation, and then for part D, it wants the cost if you produce 10,000 copies. So basically, you're finding the cost when the number of copies is 10,000. So use this formula right here and plug in what you have. We're gonna find why we're going to 0.2 times tune 1000 then add on 2000. So why is 200 plus 2000? The total cost is 2000 200 and then for Part E. I'll put it right here. It says, How many copies will be produced? So we're gonna find X if the cost is 2600 so same formula. We're plugging in the cost and we want to solve it for X. So let's subtract 2000 from both sides. So we get 600 equals 0.2 x, and then you got divide both sides by a 0.2 So calculator time do 600 divided by 6000.2 and you should get 30 1000 that's copies.

I'm afraid you're going to program about statistics here. We have to minimize feet off X one comma x two, which is going to fight X one squared plus 25 excellent plus 0.0 fight x two squared because dual x two Subject o x one plus x two minus 1000 equals zero Defined the at X y lambda It was pointed to fight excellent square plus 25 excellent plus 0.0 fight extra square plus to elect to minus lamb. Dying do X one The 62 minus those that the excellent equals there are going to fight. Excellent. That's 25 minus lambda equals zero deep. It's just a text to you cause 0.1 x two plus too well, My last Lambda equals zero The transport to Lambda equals x one plus x two. My last 1000 equals zero. If we simplify, we get I've excellent minus x two plus 1 30 equals zero x one plus x two minus 1000 equals zero. It has a solution off. 1 45 Coma. 8 55. Thank you


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